I learned to have good vibrato technique from Dean Parks (who has played with practically everybody and their brother). His approach did wonders for my playing and now I'd like to pass on those lessons to you.
We're showing this video from a player's eye-view so you can get a sense of how vibrato comes from the wrist. Start with a half-bend (I'm playing G# to A on the second string, if you want to play along), and use your wrist to push the note up and back, up and back. If you've been gripping the neck with all your strength, you'll definitely need to let up.
Vibrato comes naturally when you're already bending a note, as with the half-step vibrato we just tried. If you're sitting on a note without bending, there's less pitch change available in the vibrato but it still all comes from the wrist.